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THE VALUE DISTRIBUTION OF INCOMPLETE GAUSS SUMS

Part of:Exponential sums and character sums

Published online by Cambridge University Press:  19 February 2013

Emek Demirci Akarsu and
Jens Marklof
Emek Demirci Akarsu*
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, U.K.
Jens Marklof*
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, U.K.
*
E.DemirciAkarsu@bristol.ac.uk
j.marklof@bristol.ac.uk

Abstract

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It is well known that the classical Gauss sum, normalized by the square-root number of terms, takes only finitely many values. If one restricts the range of summation to a subinterval, a much richer structure emerges. We prove a limit law for the value distribution of such incomplete Gauss sums. The limit distribution is given by the distribution of a certain family of periodic functions. Our results complement Oskolkov’s pointwise bounds for incomplete Gauss sums as well as the limit theorems for quadratic Weyl sums (theta sums) due to Jurkat and van Horne and the second author.

MSC classification

Secondary:11L05: Gauss and Kloosterman sums; generalizations

Information

Type
Research Article
Information
Mathematika , Volume 59 , Issue 2 , July 2013, pp. 381 - 398
Copyright
Copyright © University College London 2013 

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